Fermatean Fuzzy Artinian and Noetherian Rings

This paper introduces and investigates the concepts of Fermatean fuzzy Artinian and Noetherian rings,extending classical ring-theoretic chain conditions into the framework of Fermatean fuzzy logic. We define Fermatean fuzzy left ideals and characterize the notions of Fermatean fuzzy left Artinian and Noetherian rings via descending and ascending chain conditions, respectively. We establish several structural properties, including the behavior of Fermatean fuzzy level subsets under ring homomorphisms, and the finiteness of the image sets of membership and non-membership degrees in Fermatean fuzzy Artinian and Noetherian rings. We prove that the direct product of Fermatean fuzzy Artinian rings is also Fermatean fuzzy Artinian and we present a characterization showing that the necessary and sufficient condition for a ring to be Fermatean fuzzy Noetherian is that the value set of every Fermatean fuzzy left ideal is a well-ordered subset of the unit interval. These results contribute to the growing literature on fuzzy algebra and demonstrate the usefulness of Fermatean fuzzy logic in generalizing and enriching classical algebraic concepts.